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Number of closed paths of length n whose steps are 20th roots of unity, U_20(n).
2

%I #23 Apr 30 2024 04:04:25

%S 1,0,20,0,1140,480,102800,151200,12310900,38707200,1812247920,

%T 9574488000,313983978000,2391608419200,62051403928800,611744666332800,

%U 13627749414064500,160896284989440000,3253345101771050000,43527416858084016000,829176006298475046640

%N Number of closed paths of length n whose steps are 20th roots of unity, U_20(n).

%C U_20(n) (comment in article) : For each m >= 1, the sequence (U_m(N)), N >= 0 is P-recursive but is not algebraic when m > 2.

%H Andrew Howroyd, <a href="/A198800/b198800.txt">Table of n, a(n) for n = 0..200</a>

%H Gilbert Labelle and Annie Lacasse, <a href="https://doi.org/10.46298/dmtcs.2937">Closed paths whose steps are roots of unity</a>, in FPSAC 2011, Reykjavik, Iceland DMTCS proc. AO, 2011, 599-610.

%F E.g.f.: g(x)^2 where g(x) is the e.g.f. of A070190. - _Andrew Howroyd_, Nov 01 2018

%F a(n) ~ 2^(2*n) * 5^(n+3) / (Pi^4 * n^4). - _Vaclav Kotesovec_, Apr 30 2024

%o (PARI) seq(n)={Vec(serlaplace(sum(k=0, n, if(k,2,1)*(x^k*besseli(k, 2*x + O(x^(n-k+1)))/k!)^5)^2))} \\ _Andrew Howroyd_, Nov 01 2018

%Y Cf. A070190, A198808.

%K nonn

%O 0,3

%A _Simon Plouffe_, Oct 30 2011